Gaussian curvature and mean curvature sufficient to characterize a surface? Is the knowledge of Gaussian and mean curvature (and thus of the principal curvatures) sufficient to characterize a surface uniquely?
If not, is there another geometric quantity one can add to obtain a unique characterization?
 A: The Bonnet theorem (see also here) states, roughly speaking, that if we are given the data $\tilde{I}$ and $\tilde{II}$ that are proposed to be the first and the second fundamental forms of the surface, then provided they satisfy the equations of Gauss and Codazzi (and Ricci in a more general setting) there exist a surface in the Euclidean space such that its first and second fundamental forms are $I=\tilde{I}$ and $II=\tilde{II}$ respectively. Such a surface is unique up to a rigid motion of the embracing space.
The principal curvatures $\kappa_1,\,\kappa_2$ are the eigenvalues of the shape operator $I^{-1}II$. Knowing them is equivalent to knowing the Gauss $K=\kappa_1 \cdot \kappa_2$ and the mean $H=\tfrac{1}{2}\left( \kappa_1+\kappa_2 \right)$ curvatures. This knowledge is insufficient because the problem would be underdetermined: given the eigenvalues of the matrix $I^{-1}II$, there is a lot of freedom to choose the matrices $I$ and $II$.
Proofs of the Bonnet theorem can be found in Spivak's "A Comprehensive Introduction to Differential Geometry", and, of course, in many other textbooks on Differential Geometry. An advanced reader may enjoy Werner Greub's paper "Gauss-Codazzi tensor fields and the Bonnet immersion theorem" (1977).
